${}_5{F}_4$ evaluations and a family of ${\pi }^2+{\log }^2$ identities

Abstract

We evaluate the series ${\sum }_{n\ge 1}{z}^n/\left(n^2\right(\genfrac{}{}{0ex}{}{4n}{n}\left)\right)$, equal to $-\frac{z}{4}{}_5{F}_4\left(1,1,1,\frac{4}{3},\frac{5}{3};\frac{5}{4},\frac{3}{2},\frac{7}{4},2;\frac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $\theta =j\pi /N$. Each value equals $c{\pi }^2$ plus a universal rational quadratic form in three logarithms, with $c=-\frac{1}{3}{\left(1-\frac{2j}{N}\right)}^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.

13 pages